This note shows that a utility function of a homothetic preference relation satisfying u(0) = 0 is a least concave utility function if and only if it is homogeneous of degree one. An existence result and a characterization of the least concave utility of homothetic preferences | Springer Nature Link Skip to main content Advertisement An existence result and a characterization of the least concave utility of homothetic preferences Chapter pp 123–127 Cite this chapter Save chapter View saved research Advances in Mathematical Economics Abstract This note shows that a utility function of a homothetic preference relation satisfying u (0) = 0 is a least concave utility function if and only if it is homogeneous of degree one. 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Existentialism Humanism Set Theory Utilitarianism Functional Analysis Calculus of Variations and Optimization Inequalities and Integral Operators in Mathematical Analysis Notes 1. The definition of homothetic preference is in the next section. 2. The uniqueness up to a positive affine transformation means the following fact: if both u 1 and u 2 are the least concave utility function, then there exist some a > 0 and b ∈ ℝ such that \({u}_{1} = a{u}_{2} + b\) . 3. Since Ω is convex and w is continuous, w ( Ω ) must be connected. In general, any connected subset of ℝ is convex. Therefore, we have w ( Ω ) is convex. 4. The condition P ≿ ≠ ∅ is so mild that we could not find any example in which P ≿ = ∅ , except a trivial example: P ≿ = ∅ if x ∼ y for any x , y ∈ Ω . 5. This result is partially shown in Kihlstrom and Mirman [ 3 ]. 6. It can be shown by the same argument as the proof of Proposition 3.C.1 of Mas-Colell, Whinston and Green [ 4 ]. 7. It can be verify by the same argument as Exercise 2-1 of Stokey and Lucas [ 5 ]. References Debreu, G.: Least concave utility functions. J. Math. Econ. 3 , 121–129 (1976) Article Google Scholar Kannai, Y.: The ALEP definition of complementarity and least concave utility functions. J. Econ. Editor information Editors and Affiliations Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-0041, Japan Shigeo Kusuoka ( Professor ) ( Professor ) Department of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo, 108-8345, Japan Toru Maruyama ( Professor ) ( Professor ) Rights and permissions Reprints and permissions Copyright information © 2011 Springer About this chapter Cite this chapter Hosoya, Y. (2011). An existence result and a characterization of the least concave utility of homothetic preferences. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 15. Copy shareable link to clipboard Provided by the Springer Nature SharedIt content-sharing initiative Key words least concave utility homothetic preference homogeneity of degree one Publish with us Policies and ethics Profiles Yuhki Hosoya View author profile Access this chapter Log in via an institution Subscribe and save Springer+ from €37.37 /Month Starting from 10 chapters or articles per month Access and download chapters and articles from more than 300k books and 2,500 journals Cancel anytime View plans Buy Now Chapter EUR 29.95 Price includes VAT (Indonesia) Available as PDF Read on any device Instant download Own it forever Buy Chapter eBook EUR 85.59 Price includes VAT (Indonesia) Available as EPUB and PDF Read on any device Instant download Own it forever Buy eBook Softcover Book EUR 104.99 Price excludes VAT (Indonesia) Compact, lightweight edition Free shipping worldwide - view details Buy Softcover Book Hardcover Book EUR 99.99 Price excludes VAT (Indonesia) Durable hardcover edition Free shipping worldwide - view details Buy Hardcover Book Tax calculation will be finalised at checkout Purchases are for personal use only Institutional subscriptions

For any Walrasian demand function, the Strong Axiom implies (and is implied by) rationalizability by a complete preorder. However, these equivalent conditions do not ensure the existence of a continuous utility function or complete preorder giving raise to the primitive demand. We here propose a self‐contained proof of a related fact: if the demand is homothetic and continuous, the Strong Axiom characterizes the existence of a continuous and homogeneous of degree one subjective utility function (or a continuous and homothetic complete preorder) representing the demand. Our contruction depends upon standard tools and overturns the need for ad‐hoc axioms that were used in prior published literature on the topic.